Most researchers agree that invariances are desirable in computer vision systems. However, one always has to keep in mind that this is at the expense of accuracy: By construction, all invariances inevitably discard information. The concept of morphological invariance is a good example for this trade-off and will be in the focus of this paper. Our goal is to develop a descriptor of local image structure that carries the maximally possible amount of local image information under this invariance. To fulfill this requirement, our descriptor has to encode the full ordering of the pixel intensities in the local neighbourhood. As a solution, we introduce the complete rank transform, which stores the intensity rank of every pixel in the local patch. As a proof of concept, we embed our novel descriptor in a prototypical TV-L1-type energy functional for optical flow computation, which we minimise with a traditional coarse-to-fine warping scheme. In this straightforward framework, we demonstrate that our descriptor is preferable over related features that exhibit the same invariance. Finally, we show by means of public benchmark systems that our method produces – in spite of its simplicity – results of competitive quality.